Galois field multipliers which multiply two numbers in a Galois field are used in error correction encoding and decoding hardware, such as Reed-Solomon encoders or decoders. Such encoders and decoders include many Galois field multipliers, although such multipliers necessarily include very complex hardware. Some well-known Galois field multipliers are described in Berlekamp, Algebraic Coding Theory, Academic Press, 1968, at pages 47-48. Using straightforward methods of implementation, the complexity and cost of such multipliers increase very rapidly with the number of bits per byte. For example, in GF512 (GS2.sup.9), multiplication of any two of nine-bit bytes requires implementation of the multiplication table among the 512 elements of GF512. Assuming this is done using a pair of programmable read only memories, each programmable read only memory (PROM) would receive all nine bits of the multiplicand, one of the PROMs receiving five bits of the multiplier, and the other PROM receiving the remaining four bits of the multiplier, each PROM storing the requisite multiplication table. The output of the two PROMs is added (in an exclusion OR gate) and the result is the desired product. The complexity and cost of the two PROMs makes such an implementation too ponderous or costly for many applications and therefore unsuitable.